An explicitly-correlated method to obtain ionized and electron-attached states with spin-orbit couplings effects included is formulated and implemented. This formulation uses both the left and the right eigenvectors of the equation-of-motion coupled-cluster method, derived from the linearly approximated explicitly correlated ground state coupled-cluster singles and doubles [CCSD(F12)] wavefunction. The Briet-Pauli Hamiltonian within the spin-orbit mean field (SOMF) approximation is used to describe the spin-orbit interactions. Basis set dependence of the spin-orbit splittings is analyzed with the canonical and explicitly-correlated approaches. Numerical tests for several systems show fast convergence of the explicitly-correlated results to the corresponding complete basis set limit (CBS).